# Characteristic direct factor

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristic subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is termed a characteristic direct factor if it is a characteristic subgroup as well as a direct factor.

## Examples

### Extreme examples

• Every group is a characteristic direct factor of itself.
• The trivial subgroup is a characteristic direct factor in any group.

### Subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes Follows by combining characteristicity is transitive and direct factor is transitive Suppose $H \le K \le G$ are groups such that $H$ is a characteristic direct factor of $K$ and $K$ is a characteristic direct factor of $G$. Then, $H$ is a characteristic direct factor of $G$.
trim subgroup property Yes Follows from the fact that both the property of being characteristic and the property of being a direct factor satisfy this condition. In any group $G$, both the whole group $G$ and the trivial subgroup are characteristic direct factors.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant direct factor fully invariant subgroup and a direct factor fully invariant implies characteristic characteristic direct factor not implies fully invariant characteristic direct factor|fully invariant direct factor}}
Hall direct factor Hall subgroup that is a direct factor equivalence of definitions of normal Hall subgroup shows that normal Hall subgroups are fully invariant. Characteristically complemented characteristic subgroup, Fully invariant direct factor|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic central factor characteristic subgroup as well as a central factor direct factor implies central factor follows from center is characteristic and the fact that the center is not always a direct factor |FULL LIST, MORE INFO
IA-automorphism-invariant direct factor IA-automorphism-invariant subgroup and a direct factor follows from characteristic implies IA-automorphism-invariant subgroup
IA-automorphism-balanced subgroup every IA-automorphism of the whole group restricts to an IA-automorphism of the subgroup. (via IA-automorphism-invariant direct factor) (via IA-automorphism-invariant direct factor) |FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms Characteristic AEP-subgroup, Characteristic central factor, Characteristic transitively normal subgroup, Complemented characteristic subgroup, Conjugacy-closed characteristic subgroup|FULL LIST, MORE INFO
direct factor normal subgroup with a normal complement |FULL LIST, MORE INFO